The core model for almost linear iterations

We introduce 0 | • (“zero hand-grenade”) as a sharp for an inner model with a proper class of strong cardinals. We prove the existence of the core model K in the theory “ZFC + 0 | • doesn’t exist.” Combined with work of Woodin, Steel, and earlier work of the author, this provides the last step for determining the exact consistency strength of the assumption in the statement of the 12th Delfino problem (cf. [12]).